The Dyson Hamilton-Jacobi Equation with Common Noise on the Wasserstein Space of the Real Line

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that the dimension $d$ fixed there is $1$, so that $\mathcal{P}_{2}(\mathbb{R}^{d})$ is the Wasserstein space $\mathcal{P}_{2}(\mathbb{R})$ of the real line, the identification of $\mathbb{R}$ with $\mathbb{R}^{1}$ being the one recorded in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Let $\lambda_{0}\in\mathbb{R}$ and $\beta\in\mathbb{R}$ be positive and let $\kappa\in\mathbb{R}$ be nonnegative. The set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information and the free score $\Xi_{\nu}\in T_{\nu}$ of such a measure are those of that definition. The bundle $\mathcal{V}(\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R}))$ of vector fields over $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ is that of that clause; for $\nu\in\mathcal{P}_{2}(\mathbb{R})$ the inner product $\langle\cdot,\cdot\rangle_{\nu}$ and norm $\lVert\cdot\rVert_{\nu}$ of $L^{2}(\nu;\mathbb{R})$ are those of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), and $T_{\nu}\subseteq L^{2}(\nu;\mathbb{R})$ is the tangent space; $\mathcal{S}(1)$ is the set of symmetric real $1\times1$ matrices and $\mathrm{tr}\,Y$ the trace of $Y\in\mathcal{S}(1)$; and $M_{2}$ is the second moment, finite on $\mathcal{P}_{2}(\mathbb{R})$. The letter $q$ denotes a vector field, the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i) and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (i) not being used. Put $2=1+1$, $4=2+2$ and $8=4+4$; these are positive, by claim 8 of Elementary Order Arithmetic in an Ordered Field for $2$ and by claim 3 of that lemma applied twice for $4$ and $8$, so each has a multiplicative inverse by claim 7 there, and $\tfrac{s}{n}$ denotes the product of a real number $s$ with the multiplicative inverse of $n$.

1. (The operator)(i) For $(\nu,q)\in\mathcal{V}(\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R}))$ the measure $\nu$ lies in $\mathcal{P}_{2}(\mathbb{R})$, so $M_{2}(\nu)$ is a real number, and the free score $\Xi_{\nu}$ lies in $T_{\nu}$, hence in $L^{2}(\nu;\mathbb{R})$, so that $\langle\Xi_{\nu},q\rangle_{\nu}$ is a real number. The Dyson Hamilton-Jacobi operator with common noise, with discount $\lambda_{0}$, common-noise intensity $\kappa$ and inverse temperature $\beta$, is the function

$$F:\ \mathcal{V}\bigl(\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})\bigr)\times\mathbb{R}\times\mathcal{S}(1)\to\mathbb{R},$$ $$F(\nu,r,q,Y)=\lambda_{0}\,r-\frac{\kappa}{2}\,\mathrm{tr}\,Y+\frac{1}{2}\,\lVert q\rVert_{\nu}^{2}-\frac{\beta}{4}\,\langle\Xi_{\nu},q\rangle_{\nu}-\frac{1}{8}\,M_{2}(\nu),$$

a second-order equation operator over $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$.

2. (The equation)(ii) The Dyson Hamilton-Jacobi equation with common noise is

$$\lambda_{0}\,r-\frac{\kappa}{2}\,\mathrm{tr}\,Y+\frac{1}{2}\,\lVert q\rVert_{\nu}^{2}-\frac{\beta}{4}\,\langle\Xi_{\nu},q\rangle_{\nu}=\frac{1}{8}\,M_{2}(\nu),$$

an equation in $(\nu,r,q,Y)$ with $(\nu,q)\in\mathcal{V}(\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R}))$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(1)$; equivalently, $F(\nu,r,q,Y)=0$ with $F$ the operator of clause 1. A classical solution of the equation is a classical solution of $F$ on $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$. In the viscosity sense the equation is read relative to a penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ with $\mathcal{D}_{\Sigma}=\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ and $\Sigma(\nu)=-\tfrac{\beta}{4}\,\Xi_{\nu}$, for which $F$ is the Hamilton-Jacobi operator with common noise and penalty drift of that pair with discount $\lambda_{0}$, common-noise intensity $\kappa$, control cost $1$ and running cost $\nu\mapsto\tfrac{1}{8}M_{2}(\nu)$: a viscosity solution of the equation is then a function $u:\mathcal{D}\to\mathbb{R}$ that is a viscosity solution of $F$ relative to that pair; classical sub- and supersolutions and viscosity sub- and supersolutions of the equation are read in the same way.